Correlation analysis of time series goes side by side with seasonal analysis. The goal of correlation analysis is to provide insights about the relationship between the series and its lags. This information than can be use to define and tune the parameters of forecasting models such as ARIMA, TSLM, etc.

Correation analysis is mainly based on data visualization and statistical tools.

library(TSstudio)
library(plotly)
library(feasts)
library(tsibble)

Data

In the following examples we will use again the AirPassengers series:

data("AirPassengers")

ts_plot(AirPassengers,
        slider = TRUE)

The Auto Correlation Function

The Auto Correlation Function, or ACF, is the main tool for quantifying the level of correlation between a series and its lags. This method is fairly similar (both mathematically and logically) to the Pearson correlation coefficient, but has time awernace:

\[r_{k} = \frac{\sum_{t = k+1}^{n-k}(x_{t-k} - \overline{x})(x_t-\overline{x})}{\sum_{t = 1}^{n}(x_t - \overline{x})^2}, ~ where\]

In R the acf function from the stats package is the main tool for calculating the series AC (Auto Correlation), and it suports only ts objects. We later see the ACF function from the feasts package, a wrapper of the acf function, that provides support for tsibble objects. Let’s use the acf function to calculate the AC of the AirPassengers series:

acf(AirPassengers)

The Partial Auto Correlation Function

The Partial Auto Correlation Function (PACF) is a conditional correlation between a series and lag k given the impact of the lags 1 to k-1 on the series. Likewise the pcf function, the pacf function is corresponding function for calculating the PAC (Partial Auto Correlation) of a series with its lags:

pacf(AirPassengers)

The key applications of the ACF and PACF functions are:

ACF and PACF Wrappers

The ts_cor function from provides an interactibe wrapper for the acf and pacf functions, plotting both correlations:

ts_cor(AirPassengers, lag.max = 72)

By default, the function marked the seasonal lags in red. You can use the seasonal_lags argument to mark additional seasonal lags (in case exists):

ts_cor(AirPassengers, lag.max = 72, seasonal_lags = 3)

Lags Plots

The lags plot is common method for visualizing the correlation between a series and its lags. The ts_lags function from the TSstudio package provides this functionality:

ts_lags(AirPassengers)

As the relationship between a series and its past lag looks more linear, the higher the correlation between the series and the lag. In the case of the AirPassengers series, you can see that the series has strong linear relationship with the first and seasonal (lag 12) lags, as observed before with the acf function.

The lags arguments enables to plot the series aginst a specific lags. For example, we could plot the relationship of the series with its past three seasonal lags (e.g., 12, 24, and 36):

ts_lags(AirPassengers, lags = c(12,24, 36))

Correlation Analysis with the feasts Package

So far, the tools we saw above, supports ts objects. The feasts, as we saw before, provides warppers for the stats main functions for time series analysis for tsibble objects. First, let’s convert the AirPassengers to a tsibble object:

ap_tsibble <- AirPassengers %>% as_tsibble()

The ACF and PACF functions, as their names impay, provides a warappers for the acf and pacf functions. By default, unlike the origin functions it won’t plot the output and we will have to add the autoplot function to plot the results:

ap_tsibble %>% ACF(value, lag_max = 48) %>% autoplot()

ap_tsibble %>% PACF(value, lag_max = 48) %>% autoplot()

The gg_lag provides a lag plots. The nice feature of this function that it colored the observations by their frequency units. By default it used geom line, which I found a bit confusing, so we will set the geom to point:

p <- ap_tsibble %>% gg_lag(value, geom="point")

p

We can make this plot interactive by using the ggplotly function from the plotly package:

ggplotly(p)